I got a question wrong on an Exam in Applied Electromagnetics (An engineering course)...
True or False
j^(j) has infinitely many values, all of which are real numbers...?
answer is True!
Why?

I got a question wrong on an Exam in Applied Electromagnetics (An engineering course)...
True or False
j^(j) has infinitely many values, all of which are real numbers...?
answer is True!
Why?
So what does i<sup>i</sup> mean? (I'm a mathematician, so I can't help but use i, sorry.) It means e<sup>i ln i</sup>. Okay, so what does ln i mean? It's any complex number z such that e<sup>z</sup> = i, and we can calculate it as ln i = lni + i arg i, where arg i is the angle that i makes with the positive real axis. lni = ln 1 = 0, and arg i = π/2... or 5π/2 or 9π/2 or any number of the form (4n+1)π/2, where n is any integer. So ln i = i(4n+1)π/2 for any integer, and thus:
i<sup>i</sup> = e<sup>i*i(4n+1)π/2</sup> = e<sup>(4n+1)π/2</sup>
Neat.
Very Nice. Makes perfect sense. Problem sure did get me thinkin...
Thank you much.
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